Here, the first term on the right hand side (RHS) of this equation (Eq. (REF )) represents the changes under collision as a result of the various raw moments relaxing to their corresponding equilibria at rates given in terms of the relaxation matrix \( \mathbf {\mathsf {\hat{\Lambda }}} = diag \; (\omega _0 ,\; \omega _1,\; \omega _2, \omega _3, .....,\omega _8)\) , where \(\omega _j\)  (\(j=0,1,...8\) ) are the relaxation parameters, with the changes mapped back into the velocity space via the \(\mathbf {\mathsf {T^{-1}}}\)  operator. On the other hand, the second term on the RHS of Eq. (REF ) represents the effect of the body force via the source term, which is discretized using the trapezoidal rule. By applying the standard variable transformation \(\mathbf {\overline{f}}=\mathbf {f} -\frac{1}{2} \mathbf {S}\Delta t\) , the implicitness in this term can be removed. After dropping the 'overbar' symbol to simplify notation, then Eq. (REF ) simplifies to [1]}
\(\mathbf {f} (\mathbf {x}+\mathbf {e}\Delta t, t+\Delta t)- \mathbf {f} (\mathbf {x},t) = \mathbf {\mathsf {T^{-1}}}\Big [ \mathbf {\mathsf {\hat{\Lambda }}} \; \left(\; \mathbf {m}^{eq}-\mathbf {m} \;\right) + \left(\mathbf {\mathsf {I}} - \frac{\mathbf {\mathsf {\hat{\Lambda }}}}{2}\right) \mathbf {\Phi }\Delta t \Big ].\) 
